ANNALESDEL’INSTITUT FOURIER
MANUEL VALDIVIA
The space D(U) is not Br-complete Annales de l’institut Fourier, tome 27, no 4 (1977), p. 29-43
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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Fourier, Grenoble 27, 4 (1977), 29-43.
THE SPACE ^(Q) IS NOT B.-COMPLETE by Manuel VALDIVIA
The purpose of this paper is to study certain classes of locally convex spaces which have non-complete separated quotients. Consequently, we obtain some results about B^- completeness. In particular, we prove that the L. Schwartz space ^(ii) is not B^-complete, where 0, is a non-empty open subset of R7". The vector spaces which are used here are defined on the field of the real or complex numbers K. We shorten « Haus- dorff locally convex space » to « space )>. If
00 00 product JJ L^ and by L^ the direct sum © L^. If A n=l n.==-l is an absolutely convex bounded and closed subset of L, LA is the linear hull of A with the locally convex topology which has as fundamental system of neighbourhoods of the . ( 1 ) origin the family ^—A: n = 1, 2, ...^ We say that a n sequence (x^) in L is a Cauchy (convergent) sequence in the sense of Mackey, if there exists an absolutely convex closed and bounded subset B of L, such that (x^) is a Cauchy (convergent) sequence in LB. Then it is natural to define a locally complete space L as a space in which every Cauchy sequence in the sense of Mackey is convergent in the sense of Mackey in L. If a subset M of L is such that for every 30 MANUEL VALDIVIA point x of L there exists a sequence (^) in M which converges to x in the sense of Mackey, M is called locally dense in L. L is called a dual locally complete space if L'[cr(L', L)] is locally complete. If L is topologically iso- morphic to P write L ^ P. Let Q. be a non-empty open subset of the m-dimensional euclidean space R/". Then ^(0.) and ^'(ti) are the L. Schwartz spaces with the strong topologies. By
THEOREM 1. — Let L = (E 0 F) X 0 E^. Then there is a n==l quotient of L[(r(L', L)] which is topologically isomorphic to a locally dense and non-closed subspace of JJ E^[cr(E^, E^)]. 71=1
Proof. — Since F is a Frechet space which is not Banach, there exists a closed subspace G of F such that F/G ^ co, (see [21, remark at the end of § 31, 4.(1), p. 432). Let f be the canonical mapping from F onto F/G and let J be the THE SPACE 2(0.) IS NOT By-COMPLETE 31 identity mapping on E. The tensorial product J (g) f is a topological homomorphism from E § F onto E § (F/G) ^ E § (o, (see [I], Chapter I p. 38). Further, if H, is an one-dimensional space, then
E § (o = E § (fl H,) ^ n (E § HJ ^ E^, \n=l / n=-l (see [I], Chapter I p. 46). Since E is different from CD, in E'[ oo V= (2/1,2/2, •..,yn, ...) e © E,, then n=l g{x, y) = (ui(^) + y^ u^) +^, . . ., u,(^) + 2/,, . . .). The mapping g is a weak homomorphism from TN X © E^ into IlEn. such that g^ X © E,) is locally dens^and n=:l oo \ n=l / non-closed in JJ E^ (see the proof of Theorem 2 in [7]). n=l oo If I is the identity mapping of 0 E,, then it suffices to take the quotient of L[o(L, L')]by the kernel of the mapping g ° (A ° (J ® /•) x I) in order to obtain the conclu- sion of the theorem, a e d COROLLARY 1.1. — Let (EJ be a sequence of separable infinite-dimensional Frechet spaces. Then there is a quotient of 32 MANUEL VALDIVIA 00 s X © E^, topologically isomorphic to a non-closed dense 71=1 oo subspace of JJ E^. 71=1 Proof. — Since E^ is a separable Frechet space there exists an absolutely convex compact separable subset of E^, which is total in E^. We put T= {^y)eR^:x2+y2=l}. Then ^(T) ^ s and ^(T X T) ^ 5, (see [I], Chapter II, p. 54). Since ^(T) § ^(T) ^ ^(T X T), (see [I], Chapter II, p. 84-85), we have that s S s ^ s. On the other hand, s is an infinite-dimensional Frechet space, different from co, which is not Banach. 00 Using Theorem 1 we obtain a subspace P of s X ® E^ / oo \ / ^ ^ n=i such that [s X (B E^)/P is weakly isomorphic to a dense \ 7l=l // 00 00 non-closed subspace of JJ E^. Since (/^ X ® E^)/\ / P is a T»=I \ n=l // 00 Mackey space and every subspace of ]J[ E^ is also a Mackey 71=1 space, we have that (/ s X ©°° E/,\) // P is topologically iso- \ n=l II oo morphic to a dense non-closed subspace of TT E^ q.e.d. 71=1 Note 1. — 0. G. Smoljanov proves in [6] that there is a closed subspace H in s X o/^ such that {s X o^/H is a non closed dense subspace of co. This result is a conse- quence of our Corollary 1.1, identifing o with E^, n = 1, 2, ..., and noticing that c^ ^ G). THEOREM 2. — Let (EJ fee a sequence of non-Banach 00 Frechet spaces. Then (E § F) X © E^ /ia5 a quotient which is 71=1 topologically isomorphic to a non-closed dense subspace of co. Proof. — Since E^ is not a Banach space there is a subspace G, of (E § F) X © E, lying in E,, such that EJG, ^ co. 71=1 THE SPACE ^(Q) IS NOT Bp-COMPLETE 33 Let G = © G,. Then 71=1 f(E ® F) x © E,1/G ^ (E ® F) x © (EJG,) ^ (E ® F) x (^ L n==l -i/ 71=1 and the conclusion follows straightforward from Theorem 1 applied to (E § F) X co^. q.e.d. COROLLARY 1.2. — There is a quotient of s^ which is topologically isomorphic to a non-closed dense subspace of (o. Proof. — Since s § s ^ s, it is possible to write (s 0 s) X s^ ^ s^\ Now we apply Theorem 2. q.e.d. LEMMA. — The space ^(Q) has a closed subspace topolo- gically isomorphic to s^\ Proof. — In 0, let (K^) be a sequence of compact sets such that K^ c: K^+i, n == 1, 2, . . ., being K^+i the interior 00 set of K^+i, and [_j K^ == ^2. Suppose that Ki ^ 0. 71=1 We put Ko = 0. Let R^ be a closed m-dimensional rectangle containded in K^ - K^_i, n = 1, 2, ... Let E^ and F^ be the subspaces of ^(Q) formed by the elements of ^(Q) whose supports are contained in R^ and K^, respectively. 00 We now prove that E = Q) E^ is a subspace of ^(Q). 71=1 Let V^ be an absolutely convex neighbourhood of the origin in E^. Since Ei is a subspace of the Frechet space F^ there is an absolutely convex neighbourhood U^ of the origin in F^ such that Ui n Ei == Vi. Suppose that we have constructed in F,^ an absolutely convex neighbourhood of the origin U^ such that U, n [F,^ © EJ = r(U,_i u VJ, n ^ 1, Uo = {0}, being r(U^_i u VJ the absolutely convex hull of U^_i u V^. Since F,, © E^+i is closed in F^+i and E^+i and F^ are Frechet spaces, we have that F^ © E^+i is a subspace of F^+i and therefore we can find an absolutely convex neigh- 34 MANUEL VALDIVIA bourhood V^+i of the origin in F^+i such that U^ n (F, e E^,) = r(U, u V^,). 00 / 00 \ Let U = UU^. It is easy to see that U n ( Q) E^). coin- n=l \7i=l / / 00 \ cides with the absolutely convex hull F ^ ^_j V^) of L J V^ and U n F^ coincides with U^. v ra==l / n==l Since ^(D) is the inductive limit of the sequence (FJ we have that U is a neighbourhood of the origin in ^(Q) and therefore E is a subspace of ^(ti). On the other hand, E is complete, hence E is closed in ^(Q). If P and Q are two compact subsets of Rp and R9, respectively, then ^p(R^) § ^Q(R^) ^ ^p^o(R^), (see [I], Chapter II, p. 84). If a^ and (By. are two real numbers such that ocp < (By., r = 1, 2, . . ., m, and ly. == [a,., pp], then ^i,(R1) ^ 5, (see [3], p. 176). We have ^i^(R2) ^ ^i/R1) § ^i,(R1) ^ s^s ^ s, ^xWR3) ^ ^i,xi,(R2) § ^(R1) ^ 5 § 5 ^ 5, ^xi^-xiJR") ^ ^xI,x...xI,JRm-l) § ^i.(R1) ^ ^ ^ ^ 00 and therefore E^ ^ s, hence ^) E^ ^ s0^. q.e.d. 7l==l THEOREM 3. — Given a space E, suppose that E Aa<9 a subspace G topologically isomorphic to 5(N). TAen E Aa5 a separated quotient which is not locally complete. Proof. — According to Corollary 1.2, there is in G a closed subspace H so that G/H is topologically isomorphic to a non-closed dense subspace of o. Since G/H is a closed subspace of E/H and G/H is not locally complete, we have that E/H is not locally complete, q.e.d. COROLLARY 1.3. — There is a quotient of ^{0.) wich is not locally complete. THE SPACE ^(Cl) IS NOT B^-COMPLETE 35 Proof. — It is immediate from the Lemma and Theorem 3. q.e.d. Note 2. — 0. G. Smoljanov prove in [6] that there is a quotient of ^(R1) which is topologically isomorphic to a non-closed dense subspace of co. Since ^(Q) has a quotient topologically isomorphic to ^(R1), [10], Corollary 1.3 can be obtained from the result of Smoljanov. The following result, which will be used afterwards was proved in [9] : 6) Let G be a Banach space of infinite dimension. If a space L is not dual locally complete, there is a linear mapping g of L into G, with closed graph, which is not weakly continuous. THEOREM 4. — In ^'(0.) there is a closed subspace G such that, if H is an arbitrary infinite-dimensional Banach space, there is a linear mapping from G into H, with closed graph, which is not weakly continuous. Proof. — Corollary 1.3 gives a closed subspace L of 2(0.) such that ^(ii)/L is not locally complete. If G is the ortho- gonal subspace of ^'(^) to L, then G is not dual locally complete. Now it suffices to use result b). q.e.d. We shall use the following result that we proved in [10] : c) Let G be a (Li¥)-space. If there is in G a closed subspace H such that G/H is a non complete separable metrizable space, then G'[(Ji(G', G)] is not B^-complete. We take G separable in [10], however the proof is valid also if G is non-separable. THEOREM 5. — Let (EJ be a sequence of in finite-dimensional separable Frechet spaces. Then the dual of (E § F) X ff) E^ with the Mackey topology is not B^-complete. n=l Proof. — It is direct consequence of result c) and Theorem 1. q.e.d. THEOREM 6. — Let (EJ be a sequence of non-Banach Frechet spaces. Then the dual of (E § F) X Q) E^ with the Mackey topology is not B^-complete. 71=1 36 MANUEL VALDIVIA Proof. — It follows straightforward from result c) and Theorem 2. q.e.d. In Theorem 7 we suppose that (EJ is a sequence of infinite- dimensional spaces such that there is an absolutely convex weakly compact separable subset of E^ which is total in E^, n = 1, 2, ... We can take an infinite-dimensional closed subspace M of E'[cr(E'5 E)] which contains a total compact separable subset. 00 THEOREM 7. — Let L = (E § F) X Q) E^. TTien t/iere is a 71=1 closed subspace G of L'^L', L)] a^d? a topological homo- morphism f from G into 1VP, such that f{G) is a non-closed locally dense subspace of 1VP. Proof. — Let T = E/M1, being M1 the orthogonal subspace of E to M. Then it is possible to identify T'[(T(T', T)] with M. If T^ == T, n = 1, 2, . . ., we can write T^ in the form 00 JJ T^. Let X be the weak dual of E 0 F. In the same way n==l as in the proof of Theorem 1, we see that there is a topological 00 homomorphism u from E 0 F onto JJ T^. Let ^u be the mapping from 7l=l f®T,)fo('©T;,nT^1 into X, \7i=l / L \7i=l n=i /J which is the transposed of u. Let Y == ^u (Q)°° T^\) and 7t--=l 7 Y^ === ^(T^). Obviously, ^u is an injective topological homomorphism and therefore Y and Y^ n == 1, 2, ..., are closed subspaces of X which can be identified with the weak duals of JJ T^ and T^ respectively. 71 ==1 Taking into account result a) being G == E^[(Ji(E^ EJ] and H = Y^, we can find an injective continuous mapping ^ from E^[(T(E^ EJ] into Y^ such that ^(E^) is a non- closed dense subspace of Y^. We define a mapping f from G^Yxn^M^G")] into n^ ;l--=l 7l=i THE SPACE ^(Q.) IS NOT B^-COMPLETE 37 such that if 00 V= S i/neY, y,eY«, n = 1, 2, ..., 7»==1 00 with ^ 2/n having only a finite number of non-zero terms and "=1 00 x= (x^x^ ...,^, ...) ell^ ,, "=i then f{y. ^ = (2/1 + ^l), 2/2 + ^2), . . ., Vn + ^), . . .). The mapping f is a topological homomorphism from G 00 into U Y^ ^ MN such that its image is a non-closed locally 7l==l oo dense subset of H Y^ (see the proof of Theorem 1). q.e.d. n==l THEOREM 8. ~ Let (EJ be a sequence of infinite-dimensional separable Frechet spaces. Let L = (E § F) X ® E^. Then there is a separated quotient of L^L', L)] which is not locally complete. Proof. -— In E^ there is an absolutely convex total sepa- rable weakly compact subset, and therefore it is possible to apply Theorem 7. With the same notation as in Theorem 7, let B be the kernel of f. Then G/B is a closed subspace of L'l^L', L)]/B. Since G/B is not locally complete, we can conclude that L'l^L', L)]/B is not locally complete, q.e.d. COROLLARY 1.8. — Let H be a space which contains a subspace M topologically isomorphic to s^\ Then there is a separated quotient of H' [(Ji(H', H)] which is not locally complete. Proof.—Let M1 be the subspace of H'[pi(H', H)] ortho- gonal to M. Let 9 be the canonical mapping from H'[(i(H', H)] onto H'[(I(H', H)]/M1. Since s^ ^ s X s^ ^ {s § s) X s^, 38 MANUEL VALDIVIA we apply Theorem 8 and obtain a separated quotient P of H'[[ji(H', H)]/M1 which is not locally complete. Let ^ be the canonical mapping from H'l^H', H)][M-1- onto P. If Q is the kernel of ^ o THEOREM 9. — There is a closed subspace G of Qi{Q) such that, if L is an arbitrary infinite dimensional Banach space, it is possible to find a linear mapping from G into L, with closed graph, which is not weakly continuous. Proof. — Because of Corollary 1.8 and the Lemma, there is a closed subspace H of ^'(0.) so that ^'(Q)/H is not locally complete. If G is the orthogonal subspace of Q>(£1} to H, then G is not dual locally complete. It is suffices to apply result b). q.e.d. In order to prove Theorem 10 and Theorem 12 we shall need the following result that we have proved in [10] : d) Let E be a separable space. Let (E^) be an increasing sequence of subspaces of E so that E = [_j E^. If there is a bounded m=l set A in E such that A ^ E^, m = i, 2, . . ., there is a dense subspace F of E, F ^ E, so that F n E^ is finite- dimensional for e^ery m positive integer. THEOREM 10. — Let (EJ be a sequence of in finite-dimensional 00 Frechet spaces. Then (E § F) X ® E^ is not B^-complete. 71=1 Proof. — Since every closed subspace of a B^-complete spaces is B^-complete, [4], we can suppose that E, F, E^, n = 1, 2, . . ., are separable spaces. With the same notation as in the proof of Theorem 7, let B THE SPACE Q(Q.) IS NOT B^-COMPLETE 39 be the kernel of /*. Then B is a closed subspace of P = X x ft E;[ 71 ==1 which lies in G. Let (LJJ be a fundamental system of neighbourhoods of the origin in E § F such that U^ ^ U^+i, n = 1, 2, ... Being A^ the polar set of U^ in X, let H^ be the linear hull of A^ with the induced topology of (?(X, E § F) and let G.=H,xnE.[ 7»=1 Let z^ be an element in Y^ such that z^ f ^(E^). Then the nth coordinate of f {z^ 0) coincides with z^ and the remaining are equal to zero, hence (/* (z^ 0)) is a bounded 00 sequence in U Y^. 71=1 If h is the canonical mapping from P onto P/B, h coin- cides with f on G and consequently {h{z^ 0)) is a bounded sequence in P/B. XA is a Banach space with Y r\ XA^, as a closed subspace. Since / P \ SY. nXA, \w=l / is a closed subset of XA^ and uii Y^:p=l,2, ...i =Y, (m=l ) we can use the Baire theorem and we obtain a positive integer r such that (S Y^nX^=YnX^. \w=l / Therefore, if q is an integer larger than r Y, n XA, = {0}, and thus Zq ^ H^. Now fix q > r and assume that for some n there exists an element ^ in P such that h{Q = /^, 0) 40 MANUEL VALDIVIA with ^ == (2/n? (^1, ^2, .. ., ^ •• •))? 00 ^ e H,,, (a-i, a;2, . . ., x^ ...} = xe^[ E,,. m=l Therefore h(t^ — ^, 0)) = 0 and, since (^, 0) e G and B lies in G, it results that ^ e G. Thus y^ e Y. Hence ^ e Y n H, = Y n X^ = (^ Y,) n X^. \m=l / We can write Vn = S ?/np, 2/np ^ Y^ p=i and therefore ^n) = /'(^/^ ^) == (yn, + ^l(^l), 2/n, + ^(^ • • ^ 2/n, + ^(^r), ^r+l^r+l), ^+2(^+2), • • •) and thus the qth component in h(t^) coincides with ^q{Xq). On the other hand, h{z^ 0) = f {Zq, 0) has its qth component equal to Zq and since Zq f ^(E^) this a contradiction. Consequently h{zq, 0) ^ /^(GJ. From the last considerations we can assert that the subset of P/B {h{z,, 0), h(z^ 0), ...,A(z,,0), ...} is bounded and is not included in A(GJ for any positive integer n. Moreover, P/B- u {A(GJ: n=l,2, ...}. We can apply result rf) in order to obtain a dense non closed subspace V in P/B such that V n A(GJ is finite-dimen- sional, n = 1, 2, . . . If XQ is a point lying in the closure of G^ n /r^V) in G^, then h(xo) is a point of the closure of A(GJ n V in A(GJ. The space A(GJ n V is finite-dimen- sional and thus h(xo) E A(GJ n V. Hence XQ e G^ n A-^V) and therefore this space is closed in G^. If A is a compact subset of P, let A^ and A^ be the 00 projection of A onto X and ]J E^, respectively. Since (AJ n=l is a fundamental system of compact subsets in X, there is a THE SPACE ^(Q) IS NOT By-COMPLETE 41 positive integer riQ such ,that A(l) <== A^. Therefore, A e G^. Moreover G^ n /^(V) is closed in G^, thus A 0 /r'^V) is compact. Since /^(V) is a non-closed dense subspace of P 00 , it results that (E § F) X (B E^ is not By-complete. q.e.d. n=i THEOREM 11. — Let G be a space which contains a subspace L topologically isomorphic to 5(N). Then G is not 1Sj.~complete. Proof. — Since every closed subspace of a By-complete space is By-complete, [4], and L ^ ^N) ^ s X s^ ^ {s § s) X s^\ it suffices to apply Theorem 10. q.e.d. COROLLARY 1.11. — The space ^(0.) is not B^-complete. Proof. — It suffices to apply the Lemma and Theorem 11. q.e.d. THEOREM 12. — Let (EJ be a sequence of infinite-dimen- sional separable Frechet spaces. Let G be a separable countable inductile limit of Frechet spaces. If there is a subspace L of G 00 which is topologically isomorphic to (E § F), X ® E^, then G'[(i(G', G)] is notB^complete. Proof. — Let us suppose that G is the inductive limit of the increasing sequence (GJ of Frechet spaces. From Theorem 1 it follows that there is a closed subspace Q of L such that L/Q is topologically isomorphic to a non-closed 00 dense subspace of ]J E^. m=l If 00 gically isomorphic to a non-closed dense subspace of JJ E^. Consequently, let (rip) be an increasing sequence of positive integers such that cp(G^) n Let us pick an element Xp^ e A = {a^i, a2^, . . ., a^, . . . } is bounded. Hence A is a bounded set in G/Q which is not contained in Note. — In [10] we proved that the space ^'(fl) is not B^-complete. This result follows also from our Theorem 12, considering that ^(^) is a countable strict inductive limit of separable Frechet spaces and that it has a subspace L such that L ^ s^ ^ (s 0 s) X s^\ THE SPACE 2(0.) IS NOT Bp-COMPLETE 43 BIBLIOGRAPHY [1] A. GROTHENDIECK, Produits tensoriels topologiques et espaces nucle- aires, Mem. Amer. Math. Soc., No. 16 (1966). [2] G. KHOTE, Topological Vector Spaces I. Berlin-Heidelberg-New York, Springer 1969. [3] A. PIETSCH, Nuclear locally convex spaces. Berlin-Heidelberg-New York, Springer 1972. [4] V. PTAK, Completeness and open mapping theorem, Bull. Soc. Math. France, 86 (1958), 41-74. [5] D. A. RAIKOV, On B-complete topological vector groups, Studia Math., 31 (1968), 295-306. [6] 0. G. SMOLJANOV, The space D is not hereditarily complete, Izv. Akad. Nauk SSSR, Ser. Math., 35 (3) (1971), 686-696; Math. USSR Izvestija, 5(3)(1971), 696-710. [7] M. VALDIVIA, On countable locally convex direct sums, Arch. d. Math., XXVI, 4 (1975), 407-413. [8] M. VALDIVIA, On B^-completeness, Ann. Inst. Fourier, Grenoble 25, 2 (1975), 235-248. [9] M. VALDIVIA, Mackey convergence and the closed graph theorem, Arch. d. Math., XXV, 6 (1974), 649-656. [10] M. VALDIVIA, The space of distributions ^'(0.) is not B^-complete, Math. Ann., 211 (1974), 145-149. Manuscrit recu Ie 27 juillet 1976 Propose par J. Dieudonne. Manuel VALDIVIA, Facultad de Ciencias Paseo Valencia al Mar, 13 Valencia (Espagne).